Multiplying and dividing algebraic fractions

Algebra, grade 9 Lesson 14 of 26Learning

Multiplying and dividing algebraic fractions

Algebraic fractions are multiplied and divided exactly like number fractions:

Multiplying: numerator times numerator, denominator times denominator.

\displaystyle \frac{a}{b}\cdot\frac{c}{d}

\displaystyle =

\displaystyle \frac{ac}{bd}

\displaystyle b\neq0,\ d\neq0

Dividing: multiplying by the reciprocal of the second fraction.

\displaystyle \frac{a}{b}:\frac{c}{d}

\displaystyle =

\displaystyle \frac{a}{b}\cdot\frac{d}{c}

\displaystyle b\neq0,\ c\neq0,\ d\neq0

Multiplying: factor, cancel, multiply

Cancel before multiplying: the factors are still in sight, and the numbers stay small.

Factor every numerator and every denominator, cancel a factor that appears both above and below – even in different fractions – and only then multiply what is left:

\frac{x^{2}-4}{3x}\cdot\frac{6}{x+2}=\frac{(x-2)(x+2)}{3x}\cdot\frac{6}{x+2}=\frac{(x-2)\cdot2}{x}=\frac{2(x-2)}{x}

The domain: $\displaystyle x\neq0$ , $\displaystyle x\neq-2$ – the denominators of both fractions.

Exercise 1

Multiply.
Factor and cancel before you multiply.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Dividing: the reciprocal

To divide by a fraction, multiply by its reciprocal – the same fraction upside down. Then go on as in multiplying:

\frac{x+3}{x}:\frac{x^{2}-9}{x^{2}}=\frac{x+3}{x}\cdot\frac{x^{2}}{(x-3)(x+3)}=\frac{x}{x-3}

The domain of a division

Careful: find the domain before you turn the second fraction over – afterwards it is easy to miss the zeros of its numerator.

In a division, the numerator of the fraction we divide by also counts: turned over, it becomes a denominator – and dividing by a fraction equal to 0 is not allowed. In the example above:

the denominators: $\displaystyle x\neq0$ ;
the numerator of the divisor, $\displaystyle {x^{2}-9}={(x-3)(x+3)}$ : $\displaystyle x\neq3$ , $\displaystyle x\neq-3$ .

Exercise 2

Divide.
First turn the second fraction into a multiplication by its reciprocal.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Write the domain of the expressions in Exercise 2, parts c and d.

Formulas: algebra, grade 9

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The laws of powers

Multiplying powers with the same base
$\displaystyle a^{m}\cdot a^{n}$ $\displaystyle =$ $\displaystyle a^{m+n}$
Dividing powers with the same base
$\displaystyle a^{m}:a^{n}$ $\displaystyle =$ $\displaystyle a^{m-n}$ $\displaystyle a\neq0$
A power of a power
$\displaystyle \left(a^{m}\right)^{n}$ $\displaystyle =$ $\displaystyle a^{m\cdot n}$
A power of a product
$\displaystyle (a\cdot b)^{n}$ $\displaystyle =$ $\displaystyle a^{n}\cdot b^{n}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{n}$ $\displaystyle =$ $\displaystyle \frac{a^{n}}{b^{n}}$ $\displaystyle b\neq0$
The power zero
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$

The laws of square roots

The root of a product
$\displaystyle \sqrt{a\cdot b}$ $\displaystyle =$ $\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle a\ge0,\ b\ge0$
The root of a quotient
$\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle =$ $\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle a\ge0,\ b>0$
The square of a root
$\displaystyle \left(\sqrt{a}\right)^{2}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$
The root of a square
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The extended distributive law and the short multiplication formulas

The extended distributive law
$\displaystyle (a+b)(c+d)$ $\displaystyle =$ $\displaystyle ac+ad+bc+bd$
The square of a sum
$\displaystyle (a+b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}+2ab+b^{2}$
The square of a difference
$\displaystyle (a-b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}-2ab+b^{2}$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^{2}-b^{2}$

The parabola

Standard form
$\displaystyle y=ax^{2}+bx+c$
Vertex form – the vertex is $\displaystyle (b,k)$
$\displaystyle y=a(x-b)^{2}+k$
Product form – it meets the $\displaystyle x$ -axis at $\displaystyle x_1$ and $\displaystyle x_2$
$\displaystyle y=a(x-x_1)(x-x_2)$
The vertex’s $\displaystyle x$ , in standard form
$\displaystyle x=-\frac{b}{2a}$

The quadratic equation

The roots of $\displaystyle {ax^{2}+bx+c}={0}$ , when $\displaystyle a\neq0$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$